If the foci are points $p,q\in\mathbb{R}^{2}$ on a horizontal line
and a point on the ellipse is $c\in\mathbb{R}^{2}$, then the string
length $\ell=\left|p-c\right|+\left|q-c\right|$ (the distance from
the first focus to the point on the ellipse to the second focus) determines
the semi-axis lengths. Using the Pythagorean theorem, the vertical
semi-axis has length $\sqrt{\frac{\ell^{2}}{4}-\frac{\left|p-q\right|^{2}}{4}}$.
Using the fact that the horizontal semi-axis is along the line joining
$p$ to $q$, the horizontal semi-axis has length $\frac{\ell}{2}$.
Thus the area is $\pi\sqrt{\frac{\ell^{2}-\left|p-q\right|^{2}}{4}}\frac{\ell}{2}$
($\pi$ times each semi-major axis length, analogous to the circle
area formula $\pi r^{2}$).